Calculate the buckling stress σkrit = π²E/λ², i.e. the compressive stress present in the member when it reaches the Euler buckling load. It is the fastest validity test of Euler theory: if σkrit exceeds the material's proportional limit, the member no longer buckles elastically and the Euler formula overestimates the capacity.
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Calculate the buckling stress σkrit = π²E/λ², i.e. the compressive stress present in the member when it reaches the Euler buckling load. It is the fastest validity test of Euler theory: if σkrit exceeds the material's proportional limit, the member no longer buckles elastically and the Euler formula overestimates the capacity.
A steel member with E = 210 GPa and λ = 89.44 has σkrit = π²·210,000 N/mm²/89.44² ≈ 259 N/mm². As a check: the same member with A = 20 cm² and Fkrit = 518 kN gives Fkrit/A = 518,154 N/2,000 mm² = 259 N/mm² – identical. Since the proportional limit of S235 is about 190 N/mm², σkrit here exceeds it: the member is too stocky for the Euler formula.
Linear-elastic homogeneous material and an ideal straight member under centric load with constant cross-section. The result is usable only while σkrit stays below the material's compressive proportional limit |σP|, i.e. for λ > λ₀ = π√(E/|σP|); below that the Euler formula is inadmissible and gives unsafely high values. Initial curvature, load eccentricity, residual stresses, inelastic buckling, torsional buckling, local plate buckling of thin-walled profiles and the safety factors required by codes are excluded.
The buckling stress σkrit is the compressive stress present in a member exactly when it reaches the Euler buckling load. It replaces the buckling-load calculation with a stress that can be compared directly against material data – answering in one step whether Euler theory applies to the member at all.
Dividing the Euler buckling load Fkrit = π²EI/lk² by the cross-sectional area A and replacing I/A with the squared radius of gyration i² gives σkrit = π²E/λ² with the slenderness ratio λ = lk/i. The geometry is thus fully contained in one dimensionless number, leaving only the elastic modulus. Plotted against λ, σkrit is the Euler hyperbola: the right-hand, elastic branch of every buckling curve.
Like tyre pressure: the force alone says little; only pressure – force per area – can be compared against a rated limit. σkrit turns the buckling load into exactly such a comparable quantity that can be placed next to the material's yield or proportional limit.
σkrit = π² · E / λ²
Buckling stress: σkrit = π²·E/λ²Derivation: σkrit = Fkrit/A = π²EI/(lk²A) = π²E/λ²Slenderness ratio: λ = lk/i with i = √(I/A)Validity limit: σkrit ≤ |σP| or λ ≥ λ₀ = π·√(E/|σP|)| Symbol / input | Meaning |
|---|---|
| Buckling stress σkrit | The compressive stress in the member when the buckling load is reached, i.e. σkrit = Fkrit/A. It is a computed comparison value, not a material strength: a member whose σkrit lies above the proportional limit never reaches that value because the material yields first. |
| Elastic modulus E | Elastic modulus of the member material from a material table: structural steel ≈ 210 GPa, aluminium alloys ≈ 70 GPa, grey cast iron ≈ 100 GPa. Buckling stress depends only on E and slenderness – not on yield strength. |
| Slenderness ratio λ | The dimensionless slenderness ratio λ = lk/i with buckling length lk = β·l and radius of gyration i = √(I/A). Source: the slenderness-ratio calculator, or a profile table together with the end conditions. Because λ appears squared in the denominator, doubling slenderness leaves a quarter of the buckling stress. |
E is the material's elastic modulus from a material table (structural steel ≈ 210 GPa, aluminium ≈ 70 GPa). λ is the slenderness ratio λ = lk/i, computed from the reduced buckling length lk = β·l and the radius of gyration i = √(I/A). Both values must belong to the same member, i.e. the same end conditions and the same cross-section.
First determine λ from buckling length and section values, then compute σkrit with the elastic modulus. Then the decisive check: compare σkrit with the material's compressive proportional limit |σP|. If σkrit lies below it, the Euler calculation is admissible and you divide it by the required safety factor against buckling. If it lies above, the result is unusable – the member is too stocky and must be verified by an inelastic method or the governing code procedure.
A steel member with E = 210 GPa and λ = 89.44 gives σkrit = π²·210,000 N/mm²/89.44² = 259.1 N/mm². Cross-check via the buckling load: the same member (I = 100 cm⁴, A = 20 cm², lk = 2,000 mm) has Fkrit = 518,154 N, and 518,154 N/2,000 mm² = 259.1 N/mm² – both routes give exactly the same value. For S235 with |σP| ≈ 190 N/mm², σkrit exceeds the proportional limit, so the Euler formula is inadmissible. Extending the same member to λ = 130 lowers σkrit to 122.6 N/mm², safely inside the elastic range.
σkrit falls quadratically with slenderness: double the slenderness leaves a quarter of the stress. Plotted against λ it forms a hyperbola that runs to infinity for small λ – and that is exactly where it is physically wrong, since no material sustains arbitrarily high stress. Every buckling curve therefore cuts the Euler hyperbola off at the limit slenderness λ₀ and replaces it in the stocky range with a line or curve derived from tests.
The calculation runs in pascal internally. E may be entered in GPa or MPa, σkrit in MPa (= N/mm²), kPa, GPa or psi. λ is dimensionless and entered as a decimal, not a percentage. In hand calculations with E in N/mm², σkrit comes out directly in N/mm².
A fast validity test of Euler theory before any buckling check; comparing materials and profiles on a common stress basis; placing a member in the elastic or inelastic range of a buckling curve; plausibility-checking a buckling-load calculation via σkrit = Fkrit/A.
Valid only in the linear-elastic range, i.e. for λ > λ₀ = π√(E/|σP|). Below the limit slenderness the formula returns stresses the material never reaches, and therefore unsafe results. It further assumes an ideal straight member under centric load with constant cross-section and homogeneous material. Initial curvature and load eccentricity, residual stresses from rolling or welding, inelastic buckling, torsional buckling, local plate buckling of thin-walled profiles and code-required safety factors are not covered.
Common mistake: The most serious error is using σkrit without comparing it to the proportional limit – for stocky members the result is then dangerously high. σkrit must also not be read as a material strength or a permissible stress; it is a stability limit of the ideal member and still has to be divided by the required safety factor. And as with the buckling load: a higher yield strength does not shift σkrit, because only E enters.
A fast validity test of Euler theory before any buckling check; comparing materials and profiles on a common stress basis; placing a member in the elastic or inelastic range of a buckling curve; plausibility-checking a buckling-load calculation via σkrit = Fkrit/A.
E is the material's elastic modulus from a material table (structural steel ≈ 210 GPa, aluminium ≈ 70 GPa). λ is the slenderness ratio λ = lk/i, computed from the reduced buckling length lk = β·l and the radius of gyration i = √(I/A). Both values must belong to the same member, i.e. the same end conditions and the same cross-section.
Valid only in the linear-elastic range, i.e. for λ > λ₀ = π√(E/|σP|). Below the limit slenderness the formula returns stresses the material never reaches, and therefore unsafe results. It further assumes an ideal straight member under centric load with constant cross-section and homogeneous material. Initial curvature and load eccentricity, residual stresses from rolling or welding, inelastic buckling, torsional buckling, local plate buckling of thin-walled profiles and code-required safety factors are not covered.